Stability and bifurcation characteristics of viscoelastic microcantilevers
Mergen H. Ghayesh
Received: 28 January 2018 / Accepted: 20 March 2018
Ó Springer-Verlag GmbH Germany, part of Springer Nature 2018
The stability and bifurcations of viscoelastic microcantilevers is investigated via the Kelvin–Voigt scheme and the
modiﬁed couple stress (MCS) theory. All the nonlinearities due to large deformations (due to the movement of the free
end) are taken into account. The viscous segments of the deviatoric segment of the symmetric couple stress tensor and the
stress tensor itself are considered. The energy loss due to viscosity is balanced with the energy input to the microcantilever.
The equations for the transverse and axial motions are obtained and the inextensibility condition is applied, yielding an
integro-partial-differential equation with inertial and stiffness nonlinearities. A continuation method is used for numerical
solutions, with highlighting the viscosity effect on the large-amplitude motions.
Many micromachines, such as airbag accelerometers,
microresonators, micro energy harvesters, microswitches,
mass-ﬂow sensors, and pressure sensors work based on the
vibrations or deformations of micro scale structures such as
microbeams, microplates and micro disks and rings (Hari
et al. 2017; Menon et al. 2017; Liu et al. 2017; Lotﬁ et al.
2017; Shi et al. 2017; Samaali and Najar 2017; Saxena
et al. 2017; Li et al. 2017; Wang et al. 2017; Gaafar and
Zarog 2017; Hamzah et al. 2017; Chorsi and Chorsi 2017;
Ma et al. 2017; Farokhi et al. 2017; Ghayesh et al.
2013c, 2016a; Farokhi and Ghayesh 2018a; Chorsi et al.
2017a). Among microbeams, one class is microcantilevers
for which one end is ﬁxed and the other one is free to
deform/oscillate. Being free at one end makes this class
oscillate with large amplitudes; curvature-related nonlin-
earities are present.
Experimental investigations (Bethe et al. 1990; Teh and
Lin 1999; Elwenspoek and Jansen 2004; Tuck et al. 2005)
showed that the effect of viscosity on the mechanical
behaviour of microstructures is signiﬁcant in many cases.
Theoretically, there are different schemes such as linear
standard, Kelvin–Voigt, and Maxwell to model the
viscosity in a structure. The Kelvin–Voigt scheme is used
in this study (Ghayesh et al. 2016b).
Being small (Farokhi et al. 2013; Gholipour et al. 2015;
Ghayesh et al. 2017a) is another characteristic of
microstructures, which induces additional stiffness, where
depending on the application, can enhance or lessen the
performance. Modiﬁed continuum mechanics (Ghayesh
et al. 2013a, b, 2017c; Farokhi and Ghayesh 2017a) such as
the modiﬁed couple stress theory incorporates the addi-
Even though there are many published investigations on
both-end supported microbeams available, the published
literature on cantilevered microbeams, where the source of
nonlinearity is large curvatures, is limited (Farokhi and
Ghayesh 2018b; Ghayesh et al. 2017b; Farokhi and
Ghayesh 2016). Rahaeifard et al. (2011), for example, used
the MCST and incorporated small-size inﬂuences for a
static pull-in phenomenon analysis. Baghani et al. (2012
examined the size-dependent pull-in phenomenon in can-
tilever MEMS devices. Rokni et al. (2015) analysed the
transverse (by neglecting the axial displacement) vibrations
of a functionally graded cantilevered microbeam.
This paper is the ﬁrst to analyse the vibrations of vis-
coelastic microcantilevers subject to large-amplitude
motions via a fully nonlinear model. In the frame work of the
Kelvin–Voigt scheme (Ghayesh et al. 2018), the viscosity in
the stress tensor and the deviatoric segment of the symmetric
couple stress tensor is modelled. Curvature nonlinearities
are taken into account and a high-dimensional discretised
model is simulated. Hamilton’s method is used for an energy
input/loss balance. A continuation method is employed for
& Mergen H. Ghayesh
School of Mechanical Engineering, University of Adelaide,
Adelaide, SA 5005, Australia